The solution isn’t as simple as you might think… As is always the case with a question like this, you’ve got to be sure to check the ground truth of the place. In this case, if you go to Wailea beach on Google Maps or Google Earth, you’ll quickly find out that there’s a BIG mountain to the east of the beach. That’s Haleakala, and at 10,023 ft (3,055 m), it could cast an early morning shadow on the western side of Maui.
Guess what? It does.
So solving the challenge isn’t as easy as just doing a query for:
[ sunrise Wailea Beach ]
or using one of the sunrise calculators that are out there. In this case, what I did was the following.
(Note to the reader: Even if you don’t care for the math part, follow along. I’ll show you how make even the math really simple.)
1. First, check to see if we need to take the mountains into account.
Haleakalā is big, it’s REALLY big. Here, I used Google Earth to fly down to Wailea Beach and look due east to quickly check and see if I need to worry about this.
... and a real-life shot (with buildings and trees) to give a sense of perspective.
That massif means the sun won’t actually hit the beach until it comes up over the mountain ridgetop! So, to solve this question really means figuring out when the sun will clear the top of the mountain. And to figure THAT out, we need to figure out the angle from the horizon to the ridge of the mountain. (Because once we know THAT angle, we can figure out how long it will take the sun to climb above the ridgeline.)
Here’s a diagram of what I mean. To figure out the angle Θ we need to know the height of the mountain if you look due east from the beach and the distance from the beach to there. In other words, we need to know the elevation and distance of the mountaintop to the beach.
2. How do we get the elevation and distance information?
There are at least three ways.
(A) Use Google Maps and Terrain View
(B) Use Google Earth and "Elevation Profile"
(C) Use Google Earth's "Show Sunlight" Tool.
Here's how they work.
A. Use Google Maps and Terrain View. One way to get the data we need is to draw a line due east from Wailea Beach and then look on Google Maps in Terrain view to find the tallest point on that line by reading the contour elevations.
Here, I’ve gone to Wailea Beach in Google Maps, then used My Maps (and then “Create a Map”) to draw a single line from Wailea Beach eastward.
I then switched into Terrain view on Maps to see what the contour lines would say.
If you zoom in enough, you can see that the highest contours on this line are around 7700 feet—it’s a little hard to read on the map, but I was able to use my classical map-reading skills to read it off.
The Maps “Create a Map” told me that the distance from Wailea Beach to the ridgeline is 8.5 miles. So I could calculate the angle now.
But I want to show you ANOTHER way to get this elevation + distance information as well by using Google Earth.
B. Use Google Maps, Create a Path, then use "Show Elevation Profile." If you fly to Wailea Beach in Google Earth, you can then create a path in Earth and ask for the elevation profile along that path.
Here’s how to get this information this way...
First, on your Google Earth view of Wailea Beach, use the path tool to define a path from the beach to the top of the ridge. (It's just a straight line with two points.) I’m going to call that path “East from Wailea.” The Path tool is under the "Add" menu in Google Earth.
(Note that this is very similar to what I did in Maps.)
Now, once you’ve created the line, you can right-click (or Control-Click on a Mac) and selection “Show Elevation Profile.”
That will then create the elevation profile along the path you’ve defined. Since your path “East from Wailea” is the line that the sun will shine, you can get the elevation and distance easily—by looking at the chart, you can see the top of the ridge is 7692 feet and 8.78 miles away.
NOW we know almost everything we need. Let’s make a simple chart with our information:
We just need to figure out what Θ is. This just takes a little trig. To need to figure out the angle we need the trig relationship for that angle. YOU might remember what that identity is off the top of your head, but I wanted to be sure, so I did a quick search for:
[ trig identities ]
and found a number of nice pages that explained to me that what I needed was something like the tangent function. The tangent of an angle, I was reminded, is:
opposite / adjacent = tan ( Θ )
So, in this diagram, the side "opposite" our desired angle is just the elevation (or the height) of the mountain. To change the elevation (which I know in feet) into miles, I just convert the elevation from feet into miles using Google Convert, that is, do the query like this:
[ 7692 feet in miles ]
… or 1.45 miles. NOW I can do compute the tangent. More Google Calculator
1.45 / 8.78 = tan ( Θ )
1.45 / 8.78 = 0.1651
But remember that what I really want is the ANGLE with 0.1651 as the tangent. So back to Google Calculator, and plug in the numbers:
[ arctan (0.1651) ]
…. and we get 0.16362 -- but remember THAT’S in RADIANS!
One more Google Conversion:
[ 0.16362 radians in degrees ]
and we find that the angle we've been trying to compute all along is 9.3749 degrees.
SO NOW… to figure out when the sun will hit the beach, we have to find the sunrise time with a simple:
[ sunrise Wailea ]
and find that it rises at 7:04AM.
Okay. Now by doing a small computation, you can figure out that the sun moves about 0.27 degrees / minute in Hawai’i these days (given that the sunrise is at 7:04 and sunset is at 6:00PM).
Last step—how many minutes will it take the sun to climb 9.3749 degrees?
9.3749 / 0.27 = 34.7 minutes
So… now we have our answer.
If then sun rises at 7:04, then sun should hit the beach 34.7 minutes later, at 7:39AM, Hawai'i time.
Finale: Since I’m actually *here* in Maui, I went out to the beach this morning and took the following picture of the sun coming over the ridge at…. 7:42.
Three minutes off. I'll take that as success. I attribute it to inexactness in my drawing the line from Wailea. At that point on the ridge, the line is dropping quickly. If I measured just a bit too off the actual place (rather than due east), that would account for the 3 minute discrepancy.